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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.distribution;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.special.Erf;
import org.apache.commons.math3.util.FastMath;
/**
* This class implements the <a href="http://en.wikipedia.org/wiki/L%C3%A9vy_distribution">
* Lévy distribution</a>.
*
* @since 3.2
*/
public class LevyDistribution extends AbstractRealDistribution {
/** Serializable UID. */
private static final long serialVersionUID = 20130314L;
/** Location parameter. */
private final double mu;
/** Scale parameter. */
private final double c; // Setting this to 1 returns a cumProb of 1.0
/** Half of c (for calculations). */
private final double halfC;
/**
* Build a new instance.
*
* <p><b>Note:</b> this constructor will implicitly create an instance of {@link Well19937c} as
* random generator to be used for sampling only (see {@link #sample()} and {@link
* #sample(int)}). In case no sampling is needed for the created distribution, it is advised to
* pass {@code null} as random generator via the appropriate constructors to avoid the
* additional initialisation overhead.
*
* @param mu location parameter
* @param c scale parameter
* @since 3.4
*/
public LevyDistribution(final double mu, final double c) {
this(new Well19937c(), mu, c);
}
/**
* Creates a LevyDistribution.
*
* @param rng random generator to be used for sampling
* @param mu location
* @param c scale parameter
*/
public LevyDistribution(final RandomGenerator rng, final double mu, final double c) {
super(rng);
this.mu = mu;
this.c = c;
this.halfC = 0.5 * c;
}
/**
* {@inheritDoc}
*
* <p>From Wikipedia: The probability density function of the Lévy distribution over the
* domain is
*
* <pre>
* f(x; μ, c) = √(c / 2π) * e<sup>-c / 2 (x - μ)</sup> / (x - μ)<sup>3/2</sup>
* </pre>
*
* <p>For this distribution, {@code X}, this method returns {@code P(X < x)}. If {@code x} is
* less than location parameter μ, {@code Double.NaN} is returned, as in these cases the
* distribution is not defined.
*/
public double density(final double x) {
if (x < mu) {
return Double.NaN;
}
final double delta = x - mu;
final double f = halfC / delta;
return FastMath.sqrt(f / FastMath.PI) * FastMath.exp(-f) / delta;
}
/**
* {@inheritDoc}
*
* <p>See documentation of {@link #density(double)} for computation details.
*/
@Override
public double logDensity(double x) {
if (x < mu) {
return Double.NaN;
}
final double delta = x - mu;
final double f = halfC / delta;
return 0.5 * FastMath.log(f / FastMath.PI) - f - FastMath.log(delta);
}
/**
* {@inheritDoc}
*
* <p>From Wikipedia: the cumulative distribution function is
*
* <pre>
* f(x; u, c) = erfc (√ (c / 2 (x - u )))
* </pre>
*/
public double cumulativeProbability(final double x) {
if (x < mu) {
return Double.NaN;
}
return Erf.erfc(FastMath.sqrt(halfC / (x - mu)));
}
/** {@inheritDoc} */
@Override
public double inverseCumulativeProbability(final double p) throws OutOfRangeException {
if (p < 0.0 || p > 1.0) {
throw new OutOfRangeException(p, 0, 1);
}
final double t = Erf.erfcInv(p);
return mu + halfC / (t * t);
}
/**
* Get the scale parameter of the distribution.
*
* @return scale parameter of the distribution
*/
public double getScale() {
return c;
}
/**
* Get the location parameter of the distribution.
*
* @return location parameter of the distribution
*/
public double getLocation() {
return mu;
}
/** {@inheritDoc} */
public double getNumericalMean() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
public double getNumericalVariance() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
public double getSupportLowerBound() {
return mu;
}
/** {@inheritDoc} */
public double getSupportUpperBound() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
public boolean isSupportLowerBoundInclusive() {
// there is a division by x-mu in the computation, so density
// is not finite at lower bound, bound must be excluded
return false;
}
/** {@inheritDoc} */
public boolean isSupportUpperBoundInclusive() {
// upper bound is infinite, so it must be excluded
return false;
}
/** {@inheritDoc} */
public boolean isSupportConnected() {
return true;
}
}
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